0
SPECIAL SECTION PAPERS

State-Space Model and Kalman Filter Gain Identification by a Kalman Filter of a Kalman Filter

[+] Author and Article Information
Minh Q. Phan

Thayer School of Engineering,
Dartmouth College,
Hanover, NH 03755
e-mail: mqphan@dartmouth.edu

Francesco Vicario

Department of Acute Care Solutions,
Philips Research,
Cambridge, MA 02141
e-mail: francesco.vicario@philips.com

Richard W. Longman

Department of Mechanical Engineering,
Columbia University,
New York, NY 10027
e-mail: rwl4@columbia.edu

Raimondo Betti

Department of Civil Engineering and
Engineering Mechanics,
Columbia University,
New York, NY 10027
e-mail: betti@civil.columbia.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 12, 2017; final manuscript received June 23, 2017; published online November 8, 2017. Assoc. Editor: Prashant Mehta.

J. Dyn. Sys., Meas., Control 140(3), 030902 (Nov 08, 2017) (9 pages) Paper No: DS-17-1082; doi: 10.1115/1.4037778 History: Received February 12, 2017; Revised June 23, 2017

This paper describes an algorithm that identifies a state-space model and an associated steady-state Kalman filter gain from noise-corrupted input–output data. The model structure involves two Kalman filters where a second Kalman filter accounts for the error in the estimated residual of the first Kalman filter. Both Kalman filter gains and the system state-space model are identified simultaneously. Knowledge of the noise covariances is not required.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Juang, J.-N. , 1994, Applied System Identification, Prentice Hall, Englewood Cliffs, NJ.
Overschee, P. V. , and Moor, B. D. , 1996, Subspace Identification for Linear Systems: Theory, Implementation, Applications, Kluwer, Dordrecht, The Netherlands. [CrossRef]
Juang, J.-N. , Horta, L. G. , and Phan, M. Q. , 1992, “ System/Observer/Controller Identification Toolbox,” National Aeronautics and Space Administration, Langley, VA, Report No. TM-107566. https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19920015463.pdf
Phan, M. Q. , Horta, L. G. , Juang, J.-N. , and Longman, R. W. , 1993, “ Linear System Identification Via an Asymptotically Stable Observer,” J. Optim. Theory Appl., 79(1), pp. 59–86. [CrossRef]
Juang, J.-N. , Phan, M. Q. , Horta, L. G. , and Longman, R. W. , 1993, “ Identification of Observer/Kalman Filter Markov Parameters: Theory and Experiments,” J. Guid. Control Dyn., 16(2), pp. 320–329. [CrossRef]
Phan, M. Q. , Horta, L. G. , Juang, J.-N. , and Longman, R. W. , 1995, “ Improvement of Observer/Kalman Filter Identification (OKID) by Residual Whitening,” ASME J. Vib. Acoust., 117(2), pp. 232–238. [CrossRef]
Lin, P. , Phan, M. Q. , and Ketcham, S. A. , 2014, “ State-Space Model and Kalman Filter Gain Identification by a Superspace Method,” Modeling, Simulation, and Optimization of Complex Processes, H. G. Bock , H. X. Phu , R. Rannacher , and J. R. Schloeder , eds., Springer, Cham, Switzerland, pp. 121–132. [CrossRef]
Majji, M. , Juang, J.-N. , and Junkins, J. L. , 2010, “ Observer/Kalman-Filter Time-Varying System Identification,” J. Guid. Control Dyn., 33(3), pp. 887–900. [CrossRef]
Aitken, J. M. , and Clarke, T. , 2012, “ Observer/Kalman Filter Identification With Wavelets,” IEEE Trans. Signal Process., 60(7), pp. 3476–3485. [CrossRef]
Chang, M. , and Pakzad, S. N. , 2014, “ Observer Kalman Filter Identification for Output-Only Systems Using Interactive Structural Modal Identification Toolsuite (SMIT),” J. Bridge Eng., 19(5), p. 04014002. [CrossRef]
Vicario, F. , Phan, M. Q. , Betti, R. , and Longman, R. W. , 2015, “ Output-Only Observer/Kalman Filter Identification (O3KID),” Struct. Control Health Monit., 22(5), pp. 847–872. [CrossRef]
Overschee, P. V. , and Moor, B. D. , 1994, “ N4SID: Subspace Algorithms for the Identification of Combined Deterministic Stochastic Systems,” Automatica, 30(1), pp. 75–93. [CrossRef]
Qin, S. J. , 2006, “ An Overview of Subspace Identification,” Comput. Chem. Eng., 30(10–12), pp. 1502–1513. [CrossRef]
Chiuso, A. , 2007, “ The Role of Vector Autoregressive Modeling in Predictor Based Subspace Identification,” Automatica, 43(6), pp. 1034–1048. [CrossRef]
Qin, S. J. , and Ljung, L. , 2003, “ Closed-Loop Subspace Identification With Innovation Estimation,” 13th IFAC Symposium on System Identification, Rotterdam, The Netherlands, Aug. 27–29, pp. 887–892.
Lin, W. , Qin, S. J. , and Ljung, L. , 2004, “ On Consistency of Closed-Loop Subspace Identification With Innovation Estimation,” 43rd IEEE Conference on Decision and Control (CDC), Nassau, Bahamas, Dec. 14–17, pp. 2195–2200.
Juang, J.-N. , and Pappa, R. S. , 1985, “ An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction,” J. Guid. Control Dyn., 8(5), pp. 620–627. [CrossRef]
Juang, J.-N. , Cooper, J. E. , and Wright, J. R. , 1988, “ An Eigensystem Realization Algorithm Using Data Correlations (ERA/DC) for Modal Parameter Identification,” J. Control Theory Adv. Technol., 4(1), pp. 5–14.
Ho, B. L. , and Kalman, R. E. , 1966, “ Effective Construction of Linear State-Variable Models From Input/Output Functions,” Regelungstechnik, 14(2), pp. 545–592.
Phan, M. Q. , 2011, “ Interaction Matrices in System Identification and Control,” 15th Yale Workshop on Adaptive and Learning Systems, New Haven, CT, June 6–8.
Vicario, F. , Phan, M. Q. , Betti, R. , and Longman, R. W. , 2016, “ OKID Via Output Residuals: A Converter From Stochastic to Deterministic System Identification,” J. Guid. Control Dyn., epub.
Moor, B. D. , and Vanderwalle, J. , 1987, “ A Geometrical Strategy for the Identification of State Space Models of Linear Multivariable Systems With Singular Value Decomposition,” Third International Symposium on Applications of Multivariable System Techniques, pp. 59–69.
Phan, M. Q. , Vicario, F. , Betti, R. , and Longman, R. W. , 2017, “ Observer/Kalman Filter Identification by a Kalman Filter of a Kalman Filter,” AAS/AIAA Space Flight Mechanics Meeting, San Antonio, TX, Feb. 5–9, Paper No. AAS 17-201.
Figueiredo, E. , Park, G. , Figueiras, J. , Farrar, C. , and Worden, K. , 2009, “ Structural Health Monitoring Algorithm Comparisons Using Standard Data Sets,” Los Alamos National Laboratory, Los Alamos, NM, Report No. LA-14393.

Figures

Grahic Jump Location
Fig. 1

Lumped parameter model of a four-story building

Grahic Jump Location
Fig. 2

Input data (larger magnitude) and input noise (smaller magnitude)

Grahic Jump Location
Fig. 3

Output data (larger magnitude) and output noise (smaller magnitude)

Grahic Jump Location
Fig. 4

Initially estimated Kalman residual e(k) (larger magnitude) and the difference (smaller magnitude) between it and the theoretically optimal Kalman residual ε(k)

Grahic Jump Location
Fig. 5

System Markov parameters of identified (dots) versus truth (solid line) models, CAkB versus CrArkBr

Grahic Jump Location
Fig. 6

Kalman filter gain Markov parameters of identified (dots) versus truth (solid line) models, CAkK versus CrArkKr

Grahic Jump Location
Fig. 7

Final estimated Kalman residual e(k) (larger magnitude) from (Ar,Br,Cr,Dr,Kr) and the difference (smaller magnitude) between it and the theoretically optimal Kalman residual ε(k)

Grahic Jump Location
Fig. 8

Whiteness check for Kalman-of-Kalman residual ε(2)(k)

Grahic Jump Location
Fig. 9

Residuals of Kalman (larger magnitude) and Kalman-of-Kalman filters (smaller magnitude): ε(k) versus ε(2)(k)

Grahic Jump Location
Fig. 10

Structure for experimental example [24]

Grahic Jump Location
Fig. 11

Singular-value plot for the identification of the experimental structure in Fig. 10

Grahic Jump Location
Fig. 12

Measured (solid lines) versus predicted outputs (dots) by an identified four-output one-input model of the structure in Fig. 10

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In